We can define [[Multifunction|multifunction]], for $z\neq 0$, $\text{arg}(z) = \{ \theta\in \mathbb{R} \mid z = \lvert z \rvert e^{i\theta} \}$where $\lvert z \rvert$ denotes the [[Complex Modulus|modulus]] of $z$. The ambiguity of the argument function can be solved by defining the principal value $\text{Arg}$ of the $\arg$ function to take values in $(-\pi, \pi]$. That is for any $z \in \mathbb{C}$ we have $\text{Arg}(z)\in (-\pi, \pi]$. # Properties